Dasha Loukianova (Université d’Évry Val d’Essonne), Le 27 Mars 2020

vendredi 27 mars 2020

Résumé : We study a stochastic system of interacting neurons consisting of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to $0$ and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrtN.$ In between successive spikes, each neuron’s potential follows a deterministic flow. In a diffusive regime, as $N \to \infty$,we prove the convergence of the system to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion $W$ which is created by the central limit theorem. This Brownian motion is underlying each particle’s motion and induces a common noise factor for all neurons in the limit system. Conditionally on $W,$ the different neurons are independent in the limit system. This is the \it conditional propagation of chaos property.

Travail en collaboration avec Xavier Erny, Université d’Evry et Eva Löcherbach, Université Paris 1